We’re told the “Bourne” star was hoping that three of his daughters with wife Luciana Barroso — Isabella, 10, Gia, 7, and Stella, 5 — could join those of longtime Brooklynites Ethan Hawke and Maggie Gyllenhaal at the venerable institution.

“They had a conversation with the school, but St. Ann’s just won’t bend the rules,” said an insider. “They don’t care [who the parents of its students are]. A lot of schools will bend the rules very happily; they’ll bring celebrities’ kids in midway through the year or do whatever they want. St. Ann’s just isn’t doing it.”

St. Ann’s in Brooklyn HeightsSplash News

The school, with tuition that runs between $36,080 and $42,555 per year depending on the grade, offers classes from pre-K through high school and says on its website, “So that every child will flourish, we eschew grades, rankings and prizes in favor of ongoing dialogue and teacher reports.”

The move will be something of a homecoming for the family. When they upped sticks for the West Coast in 2013, Damon said on “Today” that it had been a tough decision to relocate.

“Basically all of our friends with little kids are out there [in LA],” he said. “We’re a little conflicted. We love it here, we’re really happy here and New York will always be here. It might just turn out to be a little jaunt out there and then a return . . . It’s hard to leave here.”